A real number that is not algebraic is called transcendental. It is easy to see that every transcendental number is irrational, but, as we see above, not every irrational number is transcendental. In 1874 Georg Cantor proved that there are only countably many algebraic numbers. Thus there must be uncountably many transcendental numbers! The vast, vast majority of real numbers are transcendental.

Despite the fact that almost every real number is transcendental, it is very difficult to prove that a given number is transcendental. Joseph Liouville discovered the first transcendental number in 1844:

In 1873 Charles Hermite proved that is transcendental and nine years later Ferdinand von Lindemann proved that is transcendental. Some other transcendental numbers are: (we have not yet proved that is transcendental), , , , and (yes, this is a real number: ).

In this sequence of three blog posts I will prove that is transcendental. At a glance the proof looks long and complicated, but the proof is really quite straightforward (this post is the longest of the three). Most of the proof is nothing more than algebraic manipulations and divisibility arguments with integers. There is some differential calculus, but nothing beyond Calculus I: the mean value theorem, the derivative of the product rule, and the limit . The proof also uses the infinitude of primes.

This proof is based on Adolf Hurwitz‘s 1893 simplification of Hermite’s proof. (To be specific, I used Herstein’s Topics in Algebra as a source for the details of Hurwitz’s proof.)